Capturing dynamics of time-varying data via topology

<p style='text-indent:20px;'>One approach to understanding complex data is study its shape through the lens of algebraic topology. While early development topological analysis focused primarily on static data, in recent years, theoretical and applied studies have turned that varies time. A time-varying collection metric spaces as formed, for example, by a moving school fish or flock birds, can contain vast amount information. There often need simplify summarize dynamic behavior. We provide an introduction summaries including vineyards [<xref ref-type="bibr" rid="b19">19</xref>], crocker plots rid="b55">55</xref>], multiparameter rank functions rid="b37">37</xref>]. then introduce new tool spaces: <i>crocker stack</i>. Crocker stacks are convenient visualization, amenable machine learning, satisfy desirable continuity property which we prove. demonstrate utility parameter identification task involving influential model biological aggregations rid="b57">57</xref>]. Altogether, aim bring broader mathematics community up-to-date spaces.</p>